Simplify and expand the following expression: $ \dfrac{3}{5x + 35}+ \dfrac{2}{2x - 16}- \dfrac{x}{x^2 - x - 56} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{3}{5x + 35} = \dfrac{3}{5(x + 7)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{2}{2x - 16} = \dfrac{2}{2(x - 8)}$ We can factor the quadratic in the third term: $ \dfrac{x}{x^2 - x - 56} = \dfrac{x}{(x + 7)(x - 8)}$ Now we have: $ \dfrac{3}{5(x + 7)}+ \dfrac{2}{2(x - 8)}- \dfrac{x}{(x + 7)(x - 8)} $ The least common multiple of the denominators is: $ 10(x + 7)(x - 8)$ In order to get the first term over $10(x + 7)(x - 8)$ , multiply by $\dfrac{2(x - 8)}{2(x - 8)}$ $ \dfrac{3}{5(x + 7)} \times \dfrac{2(x - 8)}{2(x - 8)} = \dfrac{6(x - 8)}{10(x + 7)(x - 8)} $ In order to get the second term over $10(x + 7)(x - 8)$ , multiply by $\dfrac{5(x + 7)}{5(x + 7)}$ $ \dfrac{2}{2(x - 8)} \times \dfrac{5(x + 7)}{5(x + 7)} = \dfrac{10(x + 7)}{10(x + 7)(x - 8)} $ In order to get the third term over $10(x + 7)(x - 8)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{x}{(x + 7)(x - 8)} \times \dfrac{10}{10} = \dfrac{10x}{10(x + 7)(x - 8)} $ Now we have: $ \dfrac{6(x - 8)}{10(x + 7)(x - 8)} + \dfrac{10(x + 7)}{10(x + 7)(x - 8)} - \dfrac{10x}{10(x + 7)(x - 8)} $ $ = \dfrac{ 6(x - 8) + 10(x + 7) - 10x} {10(x + 7)(x - 8)} $ Expand: $ = \dfrac{6x - 48 + 10x + 70 - 10x}{10x^2 - 10x - 560} $ $ = \dfrac{6x + 22}{10x^2 - 10x - 560}$ Simplify: $ = \dfrac{3x + 11}{5x^2 - 5x - 280}$